OFFSET
0,3
COMMENTS
G.f. = x*c(x)*c(x*c(x)) where c(x) is the generating function of the Catalan numbers C(n). Thus a(n) is the Catalan transform of the sequence C(n-1). Reference for the definition of Catalan transform is the paper by Paul Barry. - Stefan Forcey (sforcey(AT)tnstate.edu), Aug 02 2007
A129442 is an essentially identical sequence. - R. J. Mathar, Jun 13 2008
From Peter Bala, Jan 27 2020: (Start)
This sequence is the main diagonal of the lower triangular array formed by putting the sequence [0, 1, 1, 2, 5, 14, 42, ...] of Catalan numbers (with 0 prepended) in the first column (k = 0) of the array and then completing the triangle using the relation T(n,k) = T(n-1,k) + T(n,k-1) for k >= 1.
0
1 1
1 2 2
2 4 6 6
5 9 15 21 21
14 23 38 59 80 80
...
Cf. A307495.
Alternatively, the sequence can be obtained by multiplying the sequence of Catalan numbers by the array A106566. (End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
R. Bacher, On generating series of complementary plane trees arXiv:math/0409050 [math.CO], 2004. See p. 19.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5, pp. 1-24.
David Callan, A combinatorial interpretation of the Catalan transform of the Catalan numbers, arXiv:1111.0996 [math.CO], 2011.
Stefan Forcey, Convex Hull Realizations of the Multiplihedra, Theorem 3.2, p. 8, arXiv:0706.3226 [math.AT], 2007-2008.
Stefan Forcey, Aaron Lauve, and Frank Sottile, New Hopf Structures on Binary Trees, dmtcs:2740 - Discrete Mathematics & Theoretical Computer Science, January 1, 2009, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009).
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Tian-Xiao He and Louis W. Shapiro, Row sums and alternating sums of Riordan arrays, Linear Algebra and its Applications, Volume 507, 15 October 2016, Pages 77-95.
FORMULA
a(0) = 0; a(n) = C(n-1) + Sum_{i=1..n-1} a(i)*a(n-i), where C(n) = A000108(n).
G.f.: (1-sqrt(2*sqrt(1-4x)-1))/2. a(n) = (1/n)*Sum_{k=1..n} binomial(2*n-k-1,n-1)*binomial(2k-2, k-1); a(0)=0. - Stefan Forcey (sforcey(AT)tnstate.edu), Aug 02 2007
From Vaclav Kotesovec, Oct 19 2012: (Start)
D-finite with recurrence 3*(n-1)*n*a(n) = 14*(n-1)*(2*n-3)*a(n-1) - 4*(4*n-9)*(4*n-7)*a(n-2).
a(n) ~ 2^(4*n-5/2)/(sqrt(Pi)*3^(n-1/2)*n^(3/2)). (End)
G.f.: A(x) satisfies A(x)=x*(1+A(x))/((1-A(x))*(1+A(x)^3)). - Vladimir Kruchinin, Jun 01 2014
G.f. is series reversion of (x - x^2) * (1 - x + x^2) = x - 2*x^2 + 2*x^3 - x^4. - Michael Somos, Jun 01 2014
From Peter Bala, Aug 22 2024: (Start)
G.f. A(x) = 1 - 1/c(x*c(x)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
Sum_{n >= 1} a(n)*y^n = x*c(x), where y = x*(1 - x). (End)
EXAMPLE
G.f. = x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + 322*x^6 + 1348*x^7 + 5814*x^8 + ...
MAPLE
a:= proc(n) option remember; `if`(n<3, n, (14*(n-1)*(2*n-3)*a(n-1)
-4*(4*n-9)*(4*n-7)*a(n-2))/ (3*n*(n-1)))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Oct 20 2012
MATHEMATICA
a[0] = 0; a[n_] := a[n] = (2 n - 2)!/((n - 1)! n!) + Sum[ a[i]*a[n - i], {i, n - 1}]; Table[ a@n, {n, 0, 24}] (* Robert G. Wilson v, Jun 28 2007 *)
a[ n_] := If[ n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[ x - 2 x^2 + 2 x^3 - x^4, {x, 0, n}]], {x, 0, n}]]; (* Michael Somos, Jun 01 2014 *)
a[0] = 0; a[n_] := Binomial[2n-2, n-1]*Hypergeometric2F1[1/2, 1-n, 2-2n, 4] /n; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 31 2016 *)
PROG
(PARI) {a(n) = if( n<1, 0, polcoeff( serreverse( x - 2*x^2 + 2*x^3 - x^4 + x * O(x^n)), n))}; /* Michael Somos, Jun 01 2014 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Jun 24 2007
EXTENSIONS
More terms from Robert G. Wilson v, Jun 28 2007
STATUS
approved